A unified framework to generate optimized compact finite difference schemes

TL;DR

This paper introduces a unified analytical framework for deriving and optimizing compact finite difference schemes on uniform grids, ensuring spectral accuracy, stability, and flexibility to generate various scheme types.

Contribution

It presents an analytical method to determine optimal scheme coefficients by solving a spectral error minimization problem with accuracy constraints, unifying different finite difference schemes.

Findings

Optimal coefficients minimize spectral error.

Framework encompasses explicit and biased schemes.

Proven relation between derivative order and coefficient symmetry.

Abstract

A unified framework to derive optimized compact schemes for a uniform grid is presented. The optimal scheme coefficients are determined analytically by solving an optimization problem to minimize the spectral error subject to equality constraints that ensure specified order of accuracy. A rigorous stability analysis for the optimized schemes is also presented. We analytically prove the relation between order of a derivative and symmetry or skew-symmetry of the optimal coefficients approximating it. We also show that other types of schemes e.g., spatially explicit, and biased finite differences, can be generated as special cases of the framework.